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(E^(a*x)*(-(b*(-2*a + a^2*x + b^2*x)*Cos[b*x]) + (-a^2 + b^2 + a^3*x + a*b^2*x)*Sin[b*x]))/(a^2 + b^2)^2


Integrate[E^(a*x)*x*Sin[b*x], x] ==
(E^(a*x)*(-(b*(-2*a + a^2*x + b^2*x)*Cos[b*x]) + (-a^2 + b^2 + a^3*x + a*b^2*x)*Sin[b*x]))/ (a^2 + b^2)^2



           a x
Integrate[E    x Sin[b x], x] == 

  a x               2      2
(E    (-(b (-2 a + a  x + b  x) Cos[b x]) + 
 
         2    2    3        2
      (-a  + b  + a  x + a b  x) Sin[b x])) / 
 
    2    2 2
  (a  + b )

Time to compute: 0.14 second

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